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## Spectral deflation in Krylov solvers: A theory of coordinate space based methods

Venue: | ETNA |

Citations: | 4 - 1 self |

### Citations

7717 | Topics in Matrix Analysis - Horn, Johnson - 1994 |

2071 | GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
- Saad, Schultz
- 1986
(Show Context)
Citation Context ...he usual case. Treating these complications suitably is the main aim of this paper. In any case, we will see that we can first solve the restricted problem Âx =r0 by a standard method such as GMRES =-=[53]-=- and subsequently compute the still ‘missing’ component of the solution by solving a small triangular linear system. While we will quickly also review the ‘symmetric case’, where the linear system is ... |

1136 |
Methods of conjugate gradients for solving linear systems
- Hestenes, Stiefel
- 1952
(Show Context)
Citation Context ...uivalent least squares problem in coordinate space, which can be solved by updating the QR decomposition of a Hessenberg or tridiagonal matrix. Orthogonal and biorthogonal residual methods such as CG =-=[34]-=- and BICG [40, 23] can also be realized in this way, but are then normally considered less attractive, perhaps due to the possible nonexistence of some of the iterates. Here, at the end, we only intro... |

450 |
Solution of sparse indefinite systems of linear equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ...to the Galerkin condition (12.2) rn ⊥ AKn(A,r0). Other methods with the same mathematical properties are the Generalized Minimum Residual (GCR) method [17], the MINRES algorithm of Paige and Saunders =-=[49]-=- for Hermitian matrices, and, the Conjugate Residual (CR) method of Stiefel [59] for Hpd matrices. While MINRES and GMRES transplant the problem into coordinate space, CG and GCR use directly recursio... |

394 | QMR: a quasi-minimal residual method for non-Hermitian linear systems
- Freund, Nachtigal
- 1991
(Show Context)
Citation Context ... not only can annul eigenvalues, but also deflate the corresponding left and right invariant subspaces. This choice leads then in a straightforward way to a ‘truly deflated’ GMRES and to deflated QMR =-=[28]-=-. Like in the symmetric case, if Z is A–invariant, the convergence speed of the deflated method is then fully determined by the nondeflated eigenvalues of A and the corresponding invariant subspace. T... |

241 | Variational iterative methods for nonsymmetric systems of linear equations
- Eisenstat, Elman, et al.
- 1983
(Show Context)
Citation Context ...ABLE 11.1 The projections PB and QB and the projected operator ÂB for a generalization of the situation of Sections 5–8 . 1. B = I for deflated CG, BICG, and FOM [51], 2. B = A H for deflated CR, GCR =-=[17]-=-, MINRES, and GMRES, 3. B = A for deflated BICR [56]. We start here from a setting suitable for deflated BICG and BICR that will be treated fully in [30]. Then we specialize it to the setting for CG, ... |

230 |
Conjugate gradient methods for indefinite systems
- Fletcher
- 1976
(Show Context)
Citation Context ... squares problem in coordinate space, which can be solved by updating the QR decomposition of a Hessenberg or tridiagonal matrix. Orthogonal and biorthogonal residual methods such as CG [34] and BICG =-=[40, 23]-=- can also be realized in this way, but are then normally considered less attractive, perhaps due to the possible nonexistence of some of the iterates. Here, at the end, we only introduce related defla... |

214 | Solution of systems of linear equations by minimized iteration.
- Lanczos
- 1952
(Show Context)
Citation Context ... squares problem in coordinate space, which can be solved by updating the QR decomposition of a Hessenberg or tridiagonal matrix. Orthogonal and biorthogonal residual methods such as CG [34] and BICG =-=[40, 23]-=- can also be realized in this way, but are then normally considered less attractive, perhaps due to the possible nonexistence of some of the iterates. Here, at the end, we only introduce related defla... |

164 | Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices
- Freund, Gutknecht, et al.
- 1993
(Show Context)
Citation Context ... PAVn = Vn+1T n , P H A H˜ Vn = ˜ Vn+1 ˜ T n , where we may enforce that all columns ofVn+1 and ˜ Vn+1 have2-norm one, and where Dn+1 :≡ ˜ V H n+1 Vn+1 is nonsingular diagonal or, if look-ahead steps =-=[26]-=- are needed, block-diagonal. With this choice (7.6a) and (7.6b) hold. So, if we start again from the ansatz (6.5) for the approximate solutionsxn, which implies the representation (6.7) for the residu... |

115 |
Krylov subspace methods for solving large unsymmetric linear systems.
- Saad
- 1981
(Show Context)
Citation Context ... PBA = PBAQB = AQB U (B H Ũ) ⊥ ˜ Z ⊥ TABLE 11.1 The projections PB and QB and the projected operator ÂB for a generalization of the situation of Sections 5–8 . 1. B = I for deflated CG, BICG, and FOM =-=[51]-=-, 2. B = A H for deflated CR, GCR [17], MINRES, and GMRES, 3. B = A for deflated BICR [56]. We start here from a setting suitable for deflated BICG and BICR that will be treated fully in [30]. Then we... |

89 | An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations”,
- FISCHER
- 1997
(Show Context)
Citation Context ... promoted Dostál’s paper [13] to a larger audience. But, his two related papers [14, 15] are not even mentioned by her. Early citations to Mansfield, who also had two follow up papers, are by Fischer =-=[22]-=- and Kolotilina [39]. To achieve the optimality of the CG error vector in the A-norm an oblique projection has to be used (see Sections 11 and 12), which can be viewed as an A-orthogonal projection ho... |

85 | Recent computational developments in Krylov subspace methods for linear systems, - Simoncini, Szyld - 2007 |

82 |
Deflation of Conjugate Gradients with Applications to Boundary Value Problems,
- Nicolaides
- 1987
(Show Context)
Citation Context ...tely invariant subspaceZ that belongs to a set of such small eigenvalues. Ways to do that have been extensively discussed in the literature and will therefore not be a topic of this paper; see, e.g., =-=[1, 3, 6, 9, 12, 37, 42, 43, 44, 45, 48, 57, 62]-=-. By using an orthogonal projection P whose nullspace is Z the Krylov space solver is then applied only to the orthogonal complementZ ⊥ by restricting the operatorAaccordingly. The basis constructed i... |

73 | Deflated and augmented Krylov subspace techniques,
- Chapman, Saad
- 1997
(Show Context)
Citation Context ...tely invariant subspaceZ that belongs to a set of such small eigenvalues. Ways to do that have been extensively discussed in the literature and will therefore not be a topic of this paper; see, e.g., =-=[1, 3, 6, 9, 12, 37, 42, 43, 44, 45, 48, 57, 62]-=-. By using an orthogonal projection P whose nullspace is Z the Krylov space solver is then applied only to the orthogonal complementZ ⊥ by restricting the operatorAaccordingly. The basis constructed i... |

73 | Computing interior eigenvalues of large matrices
- Morgan
- 1991
(Show Context)
Citation Context ...ve missed to discuss here. First of all, the determination and choice of the approximately invariant subspaces U and Ũ or ˜ Z. This topic has been often treated in the literature; see, in particular, =-=[1, 3, 6, 9, 12, 37, 42, 43, 44, 45, 48, 57, 62]-=-; of special importance for us are [2, 3, 20] as they also determine left eigenspaces, though apply them differently later. Harmonic Ritz values (see, e.g., [42, 43]) are mostly recommended for MR met... |

70 | Restarted GMRES preconditioned by deflation,”
- Erhel, Burrage, et al.
- 1996
(Show Context)
Citation Context ...d as an A-orthogonal projection however, and has nothing to do with the oblique projections promoted here. Before, in 1992, Kharchenko and Yeremin [37], followed, in 1994, by Erhel, Burrage, and Pohl =-=[18]-=- suggested GMRES algorithms with augmented basis and a corresponding nonsingular right preconditioner that moves the small eigenvalues to a multiple large eigenvalue. Later Baglama, Calvetti, Golub, a... |

68 | Adaptively preconditioned GMRES algorithms.
- Baglama, Calvetti, et al.
- 1999
(Show Context)
Citation Context ...tely invariant subspaceZ that belongs to a set of such small eigenvalues. Ways to do that have been extensively discussed in the literature and will therefore not be a topic of this paper; see, e.g., =-=[1, 3, 6, 9, 12, 37, 42, 43, 44, 45, 48, 57, 62]-=-. By using an orthogonal projection P whose nullspace is Z the Krylov space solver is then applied only to the orthogonal complementZ ⊥ by restricting the operatorAaccordingly. The basis constructed i... |

61 | A deflated version of the conjugate gradient algorithm.
- Saad, Yeung, et al.
- 2000
(Show Context)
Citation Context ...lgorithm. The same algorithm was more than ten years later again discovered by Erhel and Guyomarc’h [19] (deflation of a previously constructed Krylov subspace), by Saad, Yeung, Erhel, and Guyomarc’h =-=[54]-=-, and, independently, by Vuik, Segal, and Meijerink [61], who combined it with preconditioning by incomplete Cholesky decomposition. All three papers refer to Nicolaides [48], but not to Dostál [13] a... |

60 | An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients
- VUIK, SEGAL, et al.
- 1999
(Show Context)
Citation Context ...er again discovered by Erhel and Guyomarc’h [19] (deflation of a previously constructed Krylov subspace), by Saad, Yeung, Erhel, and Guyomarc’h [54], and, independently, by Vuik, Segal, and Meijerink =-=[61]-=-, who combined it with preconditioning by incomplete Cholesky decomposition. All three papers refer to Nicolaides [48], but not to Dostál [13] and Mansfield [41], whose articles are much closer to the... |

56 | On the construction of deflation-based preconditioners
- Frank, Vuik
(Show Context)
Citation Context ...s of Krylov space methods with augmented basis, which was further generalized in the above mentioned survey article of Eiermann, Ernst, and Schneider [16]. Many more publications followed; see, e.g., =-=[1, 24, 45, 57, 63]-=- for further references. The starting point for the present paper has been the description of recycled MINRES or RMINRES by Wang, de Sturler, and Paulino [62], which, after a minor modification that d... |

48 | GMRES on (nearly) singular systems
- Brown, Walker
- 1997
(Show Context)
Citation Context ...hods to singular systems has been investigated in detail by Freund and Hochbruck [27, §§ 3-4] and others. In particular, the application of GMRES to such systems has been analyzed by Brown and Walker =-=[8]-=-. Lemma 2.1 of [8] adapted to our situation reads as follows. LEMMA 1. If GMRES is applied toÂx =r0 and if dim Kn = n holds for some n ≥ 1, then exactly one of the following three statements holds... |

47 | Truncation Strategies for Optimal Krylov Subspace Methods,
- Sturler
- 1996
(Show Context)
Citation Context ...ely invariant subspace Z that belongs to a set of such small eigenvalues. Ways to do that have been extensively discussed in the literature and will therefore not be a topic of this paper; see, e.g., =-=[1, 3, 6, 9, 12, 37, 42, 43, 44, 45, 48, 57, 62]-=-. By using an orthogonal projection P whose nullspace is Z the Krylov space solver is then applied only to the orthogonal complement Z ⊥ by restricting the operator A accordingly. The basis constructe... |

43 | Analysis of acceleration strategies for restarted minimum residual methods,
- Eiermann, Ernst, et al.
- 2000
(Show Context)
Citation Context ...eads to a conflict with the choice imposed by residual minimization. In contrast to our treatment, the excellent general treatment and review of augmentation methods by Eiermann, Ernst, and Schneider =-=[16]-=- is mostly restricted to the application of orthogonal projections and does not capitalize upon the knowledge of bases for bothU andZ assumed here (unless they areA–invariant and thus equal). A furthe... |

43 | Explicit preconditioning of systems of linear algebraic equations with dense matrices - Kolotilina - 1988 |

41 | Lanczos-type solvers for nonsymmetric linear systems of equations
- Gutknecht
- 1997
(Show Context)
Citation Context ...eplaced by the Petrov-Galerkin condition (12.4) rn ⊥ Kn(A H ,˜r0), with a freely selectable˜r0. There is still the drawback that iterates may not exist and further breakdown problems lurk (see, e.g., =-=[32]-=-), but this is balanced by the enormous advantage of short recurrences for iterates and residuals. Eq. (12.4) implies that the residuals rn and the so-called shadow residuals ˜rn of the fictitious lin... |

39 |
Analysis of augmented Krylov subspace methods,
- Saad
- 1997
(Show Context)
Citation Context ...ed basis but no explicit deflation of the matrix, and de Sturler [11] suggested an inner-outer GMRes/GCR algorithm with augmented basis and later, in other publications, several related methods. Saad =-=[52]-=- put together a general analysis of Krylov space methods with augmented basis, which was further generalized in the above mentioned survey article of Eiermann, Ernst, and Schneider [16]. Many more pub... |

33 |
Large-scale topology optimization using preconditioned Krylov subspace methods with recycling,”
- Wang, Sturler, et al.
- 2007
(Show Context)
Citation Context |

27 |
Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme.
- Stiefel
- 1955
(Show Context)
Citation Context ...hematical properties are the Generalized Minimum Residual (GCR) method [17], the MINRES algorithm of Paige and Saunders [49] for Hermitian matrices, and, the Conjugate Residual (CR) method of Stiefel =-=[59]-=- for Hpd matrices. While MINRES and GMRES transplant the problem into coordinate space, CG and GCR use directly recursions forxn andrn. There is an analogue family of so-called orthogonal residual (OR... |

26 | Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods
- TANG, NABBEN, et al.
(Show Context)
Citation Context ...BICGSTAB, and the one of Abdel-Rehim, Stathopoulos, and Orginos [3] for their Lanczos based combined equation and eigenvalue solver. We must also mention that in a series of papers that culminates in =-=[21, 47, 60]-=- it has been shown recently that deflation, domain decomposition, and multigrid can be viewed as instances of a common algebraic framework. Outline. We start in Section 2 by introducing the basic sett... |

23 |
Conjugate gradient method with preconditioning by projector
- Dostál
- 1988
(Show Context)
Citation Context ...on of the initial residual came from Nicolaides [48], who submitted on May 13, 1985, such a deflated CG algorithms based on the three-term recursions for iterates and residuals. Independently, Dostál =-=[13]-=- submitted in January 1987 a mathematically equivalent deflated CG algorithm based on the wellknown coupled two-term recursions; he even gave an estimate for the improvement of the condition number. I... |

22 | On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling, RIACS - Freund, Hochbruck - 1991 |

20 | Deflated iterative methods for linear equations with multiple right-hand sides
- Morgan, Wilcox
(Show Context)
Citation Context ... (QCD), the Wilson matrix A has the formA = I−κW, whereκ ∈ R andW isS-Hermitian for a diagonal matrixSwith diagonal elements ±1. See [7, 10, 29] for early contributions making use of this feature and =-=[2, 1, 46, 57]-=- for some samples of the many publications that make use of deflation in lattice QCD. So, compared to QMR, simplified QMR reduces the cost in both time and memory. Regarding modifications for the defl... |

20 | Computing and deflating eigenvalues while solving multiple right-hand side linear systems with an application to quantum chromodynamics
- Stathopoulos, Orginos
(Show Context)
Citation Context |

19 |
An augmented conjugate gradient method for solving consecutive symmetric positive definite linear systems
- Erhel, Guyomarc’h
(Show Context)
Citation Context ... for what he referred to as Nicolaides’ method of deflation, but he was actually using a 2-term CG algorithm. The same algorithm was more than ten years later again discovered by Erhel and Guyomarc’h =-=[19]-=- (deflation of a previously constructed Krylov subspace), by Saad, Yeung, Erhel, and Guyomarc’h [54], and, independently, by Vuik, Segal, and Meijerink [61], who combined it with preconditioning by in... |

17 | Lanczos-type algorithms for structured non-Hermitean eigenvalue problems
- Freund
- 1993
(Show Context)
Citation Context ...ic Lanczos algorithm since ˜ Vn = Vn and ˜ Tn = Tn = Tn. Consequently, QMR just simplifies to MINRES, where, in particular, only one matrix-vector product is needed per step. As pointed out by Freund =-=[25]-=- there are other situations where one can profit from a similar simplification. In fact, Rutishauser [50] made the point that, in theory, the matrixvector product byA H in the nonsymmetric Lanczos alg... |

17 |
Iterative Methods for Nonsymmetric Linear Systems,
- Joubert, Manteuffel
- 1990
(Show Context)
Citation Context ...Z given by [ Vn+1 Z ] . Deflated CG [48, 13, 41, 61, 19, 54] and deflated FOM are normally characterized by (12.7) rn ⊥ Kn ⊕U . For CG, i.e., for Hpd A, it has been implicitly shown in various ways =-=[13, 36, 48]-=- (see also [19, Thm. 4.1] and [54, Thm 4.2]) that this implies the following optimality result, for which we provide the sketch of a straightforward proof. THEOREM 14. Assume A is Hpd, define Kn and U... |

15 |
Nested Krylov methods based on
- Sturler
- 1996
(Show Context)
Citation Context ...ity http://etna.math.kent.edu comparison of these three preconditioners. Also in the mid-1990s, Morgan [43] proposed GMRES with augmented basis but no explicit deflation of the matrix, and de Sturler =-=[11]-=- suggested an inner-outer GMRES/GCR algorithm with augmented basis and later, in other publications, several related methods. Saad [52] put together a general analysis of Krylov space methods with aug... |

14 | Eigenvalue translation based preconditioners for the GMRES(k
- Kharchenko, Yeremin
- 1995
(Show Context)
Citation Context |

12 | A framework for deflated and augmented Krylov subspace methods
- Gaul, Gutknecht, et al.
- 2013
(Show Context)
Citation Context ... and FOM [51], 2. B = A H for deflated CR, GCR [17], MINRES, and GMRES, 3. B = A for deflated BICR [56]. We start here from a setting suitable for deflated BICG and BICR that will be treated fully in =-=[30]-=-. Then we specialize it to the setting for CG, FOM, CR, GCR, MINRES, and GMRES considered in [31], which covers most of the published approaches. Similar to the situation in our Sections 5–8 we let U ... |

12 | C.: A comparison of abstract versions of deflation, balancing and additive coarse grid correction preconditioners.
- Nabben, Vuik
- 2008
(Show Context)
Citation Context ...BICGSTAB, and the one of Abdel-Rehim, Stathopoulos, and Orginos [3] for their Lanczos based combined equation and eigenvalue solver. We must also mention that in a series of papers that culminates in =-=[21, 47, 60]-=- it has been shown recently that deflation, domain decomposition, and multigrid can be viewed as instances of a common algebraic framework. Outline. We start in Section 2 by introducing the basic sett... |

11 |
with deflated restarting
- GMRES
(Show Context)
Citation Context |

10 |
Deflation and balancing preconditioners for Krylov subspace methods applied to nonsymmetric matrices
- Erlangga, Nabben
(Show Context)
Citation Context ...ed for Ahuja’s Recycling BICG(RBICG) [4, 5], which does not fit into our framework; see Section 12 for how it relates to our work. In particular, the oblique projection applied by Erlangga and Nabben =-=[20]-=- for their version of deflated GMRES is different from our. In fact, the projection of [20] generalizes the one that is typical for deflated CG [48, 13, 41]. The connection to some of these alternativ... |

9 | Recycling BiCG with an application to model reduction
- Ahuja, Sturler, et al.
(Show Context)
Citation Context ... They are the basic tools of our approach in 2–4. But so far the oblique projectionP that is the basis of our approaches of Sections 5–9 only seems to have been used for Ahuja’s Recycling BICG(RBICG) =-=[4, 5]-=-, which does not fit into our framework; see Section 12 for how it relates to our work. In particular, the oblique projection applied by Erlangga and Nabben [20] for their version of deflated GMRES is... |

8 | Deflated and restarted symmetric Lanczos methods for eigenvalues and linear equations with multiple right-hand sides”, submitted to
- Abdel-Rehim, Morgan, et al.
(Show Context)
Citation Context |

7 |
Krylov subspace methods in lattice QCD
- Borici
- 1996
(Show Context)
Citation Context ...ion of the lattice Dirac operator in lattice Quantum Chromodynamics (QCD), the Wilson matrix A has the formA = I−κW, whereκ ∈ R andW isS-Hermitian for a diagonal matrixSwith diagonal elements ±1. See =-=[7, 10, 29]-=- for early contributions making use of this feature and [2, 1, 46, 57] for some samples of the many publications that make use of deflation in lattice QCD. So, compared to QMR, simplified QMR reduces ... |

7 |
Progress on lattice QCD algorithms
- Forcrand
- 1996
(Show Context)
Citation Context ...ion of the lattice Dirac operator in lattice Quantum Chromodynamics (QCD), the Wilson matrix A has the formA = I−κW, whereκ ∈ R andW isS-Hermitian for a diagonal matrixSwith diagonal elements ±1. See =-=[7, 10, 29]-=- for early contributions making use of this feature and [2, 1, 46, 57] for some samples of the many publications that make use of deflation in lattice QCD. So, compared to QMR, simplified QMR reduces ... |

7 |
On the use of deflation to improve the convergence of Conjugate Gradient iteration
- Mansfield
- 1988
(Show Context)
Citation Context ...1987 a mathematically equivalent deflated CG algorithm based on the wellknown coupled two-term recursions; he even gave an estimate for the improvement of the condition number. In June 1987 Mansfield =-=[41]-=- submitted additional numerical evidence for what he referred to as Nicolaides’ method of deflation, but he was actually using a 2-term CG algorithm. The same algorithm was more than ten years later a... |

7 |
restarted GMRES and Arnoldi methods for nonsymmetric systems of equations
- Implicitly
(Show Context)
Citation Context |

7 |
An extension of the Conjugate Residual Method to Nonsymmetric Linear Systems
- Sogabe, Sugihara, et al.
- 2009
(Show Context)
Citation Context ...d operator ÂB for a generalization of the situation of Sections 5–8 . 1. B = I for deflated CG, BICG, and FOM [51], 2. B = A H for deflated CR, GCR [17], MINRES, and GMRES, 3. B = A for deflated BICR =-=[56]-=-. We start here from a setting suitable for deflated BICG and BICR that will be treated fully in [30]. Then we specialize it to the setting for CG, FOM, CR, GCR, MINRES, and GMRES considered in [31], ... |

7 | Deflated BiCGStab for linear equations in QCD problems
- Abdel-Rehim, Morgan, et al.
- 2007
(Show Context)
Citation Context ...pical for deflated CG [48, 13, 41]. The connection to some of these alternative choices will be explained in Section 11. Our approach is also different from the one of Abdel-Rehim, Morgan, and Wilcox =-=[2]-=- for their deflated BiCGStab, and the one of Abdel-Rehim, Stathopoulos, and Orginos [3] for their Lanczos based combined equation and eigenvalue solver. We must also mention that in a series of papers... |

7 | By how much can residual minimization accelerate the convergence of orthogonal residual methods
- Gutknecht, Rozložník
(Show Context)
Citation Context ...hared by the class of orthogonal residual methods, whose residual norms depend in a well-known way discovered by Paige and Saunders [49] from those of the corresponding MR method; see, e.g., [16] and =-=[33]-=-.26 Conclusions. We have described several augmented and deflated Krylov methods for solving Ax = b that all fit into a common theoretical framework. They are coordinate space based in the sense that... |

6 |
Exploiting structure in Krylov subspace methods for the Wilson fermion matrix
- Frommer, Medeke
- 1997
(Show Context)
Citation Context ...ion of the lattice Dirac operator in lattice Quantum Chromodynamics (QCD), the Wilson matrix A has the formA = I−κW, whereκ ∈ R andW isS-Hermitian for a diagonal matrixSwith diagonal elements ±1. See =-=[7, 10, 29]-=- for early contributions making use of this feature and [2, 1, 46, 57] for some samples of the many publications that make use of deflation in lattice QCD. So, compared to QMR, simplified QMR reduces ... |

6 |
Beitrage zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos
- Rutishauser
- 1953
(Show Context)
Citation Context ...re, in particular, only one matrix-vector product is needed per step. As pointed out by Freund [25] there are other situations where one can profit from a similar simplification. In fact, Rutishauser =-=[50]-=- made the point that, in theory, the matrixvector product byA H in the nonsymmetric Lanczos algorithm can be avoided since, for every square matrix A there exists a nonsingular matrix S such that A T ... |

6 |
Forcrand, Progress on lattice QCD algorithms
- de
- 1996
(Show Context)
Citation Context ...he lattice Dirac operator in lattice Quantum Chromodynamics (QCD), the Wilson matrix A has the form A = I − κW, where κ ∈ R and W is S-Hermitian for a diagonal matrix S with diagonal elements ±1. See =-=[7, 10, 29]-=- for early contributions making use of this feature and [2, 1, 46, 57] for some samples of the many publications that make use of deflation in lattice QCD. So, compared to QMR, simplified QMR reduces ... |

5 | Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides
- Abdel-Rehim, Stathopoulos, et al.
- 2009
(Show Context)
Citation Context |

5 | Recycling bi-Lanczos algorithms: BiCG, CGS, and BiCGSTAB
- Ahuja
- 2009
(Show Context)
Citation Context ... They are the basic tools of our approach in 2–4. But so far the oblique projectionP that is the basis of our approaches of Sections 5–9 only seems to have been used for Ahuja’s Recycling BICG(RBICG) =-=[4, 5]-=-, which does not fit into our framework; see Section 12 for how it relates to our work. In particular, the oblique projection applied by Erlangga and Nabben [20] for their version of deflated GMRES is... |

3 |
Deflated BiCGStab for linear equations
- Abdel-Rehim, Morgan, et al.
(Show Context)
Citation Context ...pical for deflated CG [48, 13, 41]. The connection to some of these alternative choices will be explained in Section 11. Our approach is also different from the one of Abdel-Rehim, Morgan, and Wilcox =-=[2]-=- for their deflated BICGSTAB, and the one of Abdel-Rehim, Stathopoulos, and Orginos [3] for their Lanczos based combined equation and eigenvalue solver. We must also mention that in a series of papers... |

3 | On the convergence of GMRES with invariant-subspace deflation - Yeung, Tang, et al. - 2010 |

2 |
ROZLO ˇZNÍK, By how much can residual minimization accelerate the conver- Kent State University http://etna.math.kent.edu gence of orthogonal residual methods?, Numerical Algorithms
- GUTKNECHT, M
(Show Context)
Citation Context ...hared by the class of orthogonal residual methods, whose residual norms depend in a well-known way discovered by Paige and Saunders [49] from those of the corresponding MR method; see, e.g., [16] and =-=[33]-=-. Conclusions. We have described several augmented and deflated Krylov methods for solving Ax = b that all fit into a common theoretical framework. They are coordinate space based in the sense that we... |

2 | Projector preconditioning and domain decomposition methods - Dostál - 1990 |

1 | preconditioning and domain decomposition methods - Projector - 1990 |

1 |
and augmented Krylov subspace methods: Basic facts and a breakdown-free deflated
- Deflated
- 2011
(Show Context)
Citation Context ...e. Note that for n = 0 the breakdown condition r0 ∈ N( Â) can be written as N( Â)∩ K0 = {o}, in accordance with the breakdown condition for thenth step. The following simple2×2 example taken from =-=[31]-=- exemplifies a breakdown in the first step: (3.3) A :≡ [ 0 1 1 0 ] [ 1 0 , P :≡ 0 0 ] [ 0 1 , PA = 0 0 ] [ 1 , r0 :≡ 0 ] ,ETNA Kent State University http://etna.math.kent.edu 164 where Â = PAP = O an... |

1 |
deflation preconditioning of linear algebraic systems
- Twofold
- 1998
(Show Context)
Citation Context ...ll three papers refer to Nicolaides [48], but not to Dostál [13] and Mansfield [41], whose articles are much closer to their work. From a Google scholar search one can conclude that it was Kolotilina =-=[39]-=- who ultimately promoted Dostál’s paper [13] to a larger audience. But, his two related papers [14, 15] are not even mentioned by her. Early citations to Mansfield, who also had two follow up papers, ... |

1 |
Conjugate progector preconditioning for the solution of contact problems
- Dostál
- 1992
(Show Context)
Citation Context ... are much closer to their work. From a Google scholar search one can conclude that it was Kolotilina [39] who ultimately promoted Dostál’s paper [13] to a larger audience. But, his two related papers =-=[14, 15]-=- are not even mentioned by her. Early citations to Mansfield, who also had two follow up papers, are by Fischer [22] and Kolotilina [39]. To achieve the optimality of the CG error vector in the A-norm... |